Abstract
Phase synchronization occurs whenever a linearized system describing the evolution of the difference between coupled chaotic systems has at least one eigenvalue with zero real part. We illustrate numerical phase synchronization results and stability analysis for some coupled Sprott chaotic systems presented by fractional differential equations.
1. Introduction
Phase synchronization has been reported for various coupled chaotic systems [1, 2]. This phenomenon occurs when the linearized system describing the evolution of the difference between a pair of chaotic systems has some zero or positive conditional Lyapunov exponents. As we have shown in [2], this behavior also depends upon the eigenvalues of the linearized difference system. More precisely, suppose that two identical chaotic systems and are coupled, as derive and response systems, according to the method of Pecora and Carroll [3] by a continuous coupling function . If the system , which described the evolution of the difference between two identical systems, has a zero or constant solutions, then the two systems have complete synchronization or phase synchronization, recursively [2, 4–6]. Indeed, an analysis of the linearized difference system, may yield considerable information about the dynamics of the coupled chaotic systems. For the synchronization, we need to determine the conditional Lyapunov exponents of this system, and for the phase synchronization we need to also find the eigenvalues of the system [2]. As shown below, similar results apply to the phase synchronization Sprott systems [7], presented by Fractional Differential Equations (FDEs). That is, the real parts of the eigenvalues of the evolution matrix provide information about the ability to synchronize coupled chaotic systems presented by FDEs. In illustrated numerical results, we can see several cases that may arise between derive and response systems in the form of FDEs. In some cases, the difference between derive and response is constant, while in other cases it is periodic or a function of time. These and some other cases are presented in Section 2, followed by a stability discussion in Section 3.
2. Coupled Sprott Chaotic Systems Presented by FDEs
In this section we consider four different Sprott systems presented by FDEs. In each case the derive and response systems are coupled using the methods of Carroll and Pecora [3, 8].
Example 2.1. Consider the coupled Sprott-S systems presented by the FDEs: These systems are coupled through the third equation, where is the order Riemann-Liouville integral operator defined by , with and being ordinary derivative of order for time By the Grunwald-Letnikov method [9, 10] the fractional derivative is discretized as Here, is the step size, denotes the integer part of , , and are the Grunwald-Letnikov coefficients defined by These coefficients can also be evaluated, recursively, by and Using these definitions, the above coupled Sprott-S systems are discretized as follows: Numerical chaotic results for in the and planes are illustrated in Figure 1. Figure 1(a) shows the phase synchronization for , which is in complete agreement with the direct Euler solutions of the original system for Sprott-S ODEs with , and Figure 1(b) shows the phase synchronization for with the same value of h. As we can see in both figures, the trajectories of the derive and response show that the response attractor is a copy of the derive displaced by some distance in the direction. This distance depends on the initial conditions. It is easy to see that the evolution matrix in above Sprott-S systems of FDEs takes the form which has obviously a zero and two complex eigenvalues around . So we should expect the phase synchronization only between and .
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Example 2.2. In this example we consider two Sprott-C systems that are coupled by the second method of Pecora and Carroll. That is, variable in the response system is completely replaced by its counterpart variable in the derive system, In this example, for the chaotic case , the eigenvalues of the evolution matrix are −1 and zero, and hence, we expect phase synchronization between and . The numerical results from the related discretized system are illustrated in Figure 2. Figure 2(a) shows the phase synchronization between and for and that are in complete harmony with the numerical results found by Euler’s method for the coupled Sprott-C presented by ODEs. Figure 2(b) shows the phase synchronization between and for and . The difference between the derive and response in these two cases converges to a constant depending on the initial values. Indeed, as long as the phase synchronization exists, this difference between derive and response systems will remain constant for any values of and . In this case we should note that, since there is only one negative eigenvalue, the phase synchronization is very sensitive to the values of , as well as to the initial conditions. That is, a slight change in these values may replace the chaotic behavior with periodic or steady state solutions.
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Example 2.3. Next, consider two Sprott-L systems linked through the second Pecora-Carroll method: The Sprott-L system presented by ODEs with the parameters and is chaotic, and in its above coupled form, the related evolution matrix has two imaginary eigenvalues . In this case, as illustrated in Figure 3, phase synchronization between the derive and response occurs in such way that the differences between them will change in oscillatory fashion for different values of and . The frequency of this oscillation depends on the imaginary part of the eigenvalues, but its amplitude is constant depending on the initial values. This phenomenon is called marginal oscillatory synchronization [11]. Figure 3 shows solutions for , and their differences for various values of and . As illustrated in Figure 3(d), the difference between derive and response for the values is converging to zero in an oscillatory fashion. From numerical results, we note that this coupled system is no longer chaotic for values of and less than 0.96.
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Example 2.4. Finally, couple two Sprott-R systems presented by FDEs by the first method of Pecora-Carroll as follows:
This Sprott-R system is chaotic for and . Here the related evolution matrix has two zero eigenvalues. In this case, note that for , we get from the first equations in the derive and response systems. On the other hand, it is clear that from the second equations that . This means that , so , and the difference between and is a straight line with slope equal to c, while the difference between and remains constant. As illustrated in Figure 4, this is also the case for values of and less than one.
Here, as with the examples above, the behavior of the coupled systems presented by FDEs is not chaotic for values and less than 0.96. For example, Figure 5 shows the solutions of coupled Sprott-R systems for for which chaotic solutions become periodic solutions.
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3. Convergence Criteria
Suppose two identical chaotic FDEs and , as derive and response systems, are coupled according to the method of Pecora and Carroll [3] with . Then the stability analysis of linearized system , which is found by the difference between two above systems, yields a good criteria for the stability of the phase synchronization between the derive and response systems. More precisely, in the case of , it is clear from linear stability theory in dynamical systems that the stability type of the zero equilibrium in reflects the stability type of the synchronization between the two chaotic systems and depends upon the signs of the real parts of the eigenvalues A [12]. Phase synchronization also occurs if A does not have full rank, that is, if A has at least one zero eigenvalue. For the case of and less than 1, we can use well-known theorem of Matignon [13]. Because in the case of phase synchronization the error converges to a constant or remains bounded by a constant, we may modify Matignon's theorem to the following.
Theorem 3.1. Define . Then the linear system of fractional differential equations is asymptotically stable if and only if . In this case the vector converges to at the rate .
Now it is easy to see that for the coupled FDEs of Sprott-S and Sprott-L systems, in Examples 2.1 and 2.3, are 4 and , respectively. By using this modified theorem of Matignon, if there is phase synchronization in these two coupled chaotic systems, then it is convergent for any and less than one. However, for the coupled FDEs of Sprott-C and Sprott-R systems in Examples 2.2 and 2.4, on which is one, this modified theorem does not apply. However, in this case we may use the following convergence criterion which is discussed by Zhang and Sun [14] and Erjaee [12].
First we define matrix measure of as , where is the identity matrix and is any well-known matrix norm, such as one, two, infinity, or the -norm defined by with . Now, different matrix measure can be defined as with . The following theorem shows that under some conditions the phase synchronization in Sprott-C or Sprott-R is globally asymptotically stable around a constant vector on which .
Theorem 3.2. Suppose that for some matrix measure and . Then the system is globally asymptotically stable around a constant vector . Consequently there is phase synchronization between derive and response systems, which is globally asymptotically stable.
For the proof, see [12]. Now the matrix measure in coupled Sprott-C system in Example 2.2 is −1, while the matrix measure in the coupled Sprott-R system in Example 2.4 is −2. Consequently by Theorem 3.2, these two negative matrix measures guarantee the global asymptotical stability of the phase synchronizations in the coupled Sprott-C and Sprott-R systems presented by FDEs, whenever existing.
4. Conclusion
We have discussed the existence of phase synchronization in four different Sprott systems presented by FDEs. Although the chaos synchronization broadly exists in the chaotic systems, for example, refer to [15–17], phase synchronization is rear in the chaotic systems, and whenever it does exit, it is very sensitive to the fractional order of the derivatives in both derive and response systems. Since in this article we chose the two identical systems in our coupling using the method of Pecora and Carroll, we restricted ourselves to the choice of two identical values for and as the orders of derivatives in the derive and response systems. Otherwise the phase synchronization would occur for smaller values than the ones that chose here. For example, during our investigation, we saw that phase synchronization occurs for and or for even smaller values in all the above four examples. However, these systems would not be identical.
Acknowledgment
This work is supported by Qatar National Research Fund under the Grant number NPRP 08-056-1–014.